Theorem hbn1 1704

Description: Quantified Negation. Axiom C5-2 of [Monk2] p. 113. The setvar x is not free in -. A. x ph. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.) (Proof shortened by GD, 27-Jan-2018.)

Assertion

Ref	Expression
hbn1	|- ( -. A. x ph -> A. x -. A. x ph )

Proof of Theorem hbn1

Step	Hyp	Ref	Expression
1		ax-ial 1587	. 2 |- ( A. x ph -> A. x A. x ph )
2	1	hbn 1703	1 |- ( -. A. x ph -> A. x -. A. x ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie2 1547  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408

This theorem is referenced by:  hbnt  1705  modal-5  1712  dvelimfALT2  1870